\documentclass{article}
\usepackage{amsmath,geometry,verbatim}
\geometry{margin=1in}
\renewcommand{\tilde}{\widetilde}
\newcommand{\Lie}{\mathcal{L}}
\newcommand{\ura}{\mathbf}
\begin{document}

\section{Definitions}

\begin{table}[h!]
\centering
\begin{tabular}{r c}
Unit normal & $n_\alpha n^\alpha = -1$, $n_{\left[\alpha\right.} n_{\left.\beta;\gamma\right]} = 0$ \\
Acceleration & $a^\alpha = n^\alpha_{;\beta} n^\beta$ \\
Transverse metric & $\gamma_{\alpha\beta} = g_{\alpha\beta} + n_\alpha n_\beta$ \\
Extrinsic curvature & $K_{\alpha\beta} = -\frac12 \Lie_n \gamma_{\alpha\beta} $ \\
Potential vector & $A^\alpha$ \\
Electromagnetic Field & $F_{\alpha\beta} = 2 A_{[\beta;\alpha]}$ \\
Dual field & $\tilde F^{\alpha\beta} = \frac12 \varepsilon^{\alpha\beta\gamma\delta} F_{\gamma\delta}$ \\
4-current & $j^\alpha$ \\
Electric field & $E^\beta = -n_\alpha F^{\alpha\beta}$ \\
Magnetic field & $B^\beta = -n_\alpha \tilde F^{\alpha\beta}$ \\
Charge density & $\rho = -n_\alpha j^\alpha$ \\
Current & $J^\alpha = \gamma^\alpha_\beta j^\beta$
\end{tabular}
\end{table}

\subsection{3+1-formulation}

Define tangent vectors $e^\alpha_a$ and 3-metric $\gamma_{ab} = \gamma_{\alpha\beta} e^\alpha_a e^\beta_b$. Note:

\begin{gather}
g^{\alpha\beta} = \gamma^{ab} e^\alpha_a e^\beta_b \\
v_a = v_\alpha e^\alpha_a \\
v^\alpha = v^a e^\alpha_a \\
\nabla \times \ura v \Leftrightarrow -\varepsilon^{abc} v_{b;c} \\
\ura u \times \ura v \Leftrightarrow \varepsilon^{abc} u_b v_c \\
v_{\alpha;\beta} e^\alpha_a e^\beta_b = v_{a;b} \\
\dot v_a = \Lie_n v_\alpha e^\alpha_a \\
\dot v^a e^\alpha_a = \gamma^\alpha_\beta \Lie_n v^\beta
\end{gather}

\section{Field equations}

\begin{table}[h!]
\centering
\begin{tabular}{r c}
Motion of charged particle & $\dot u_\alpha = F_{\alpha\beta} j^\beta$ \\
Source equation & $F^{\alpha\beta}_{;\beta} = 4\pi j^\alpha$ \\
Sourceless equation & $\tilde F^{\alpha\beta}_{;\beta} = 0$
\end{tabular}
\end{table}

\section{Geometric relations}

\begin{align}
n^\alpha n_{\alpha;\beta}
&= \frac12 \left( n^\alpha n_{\alpha;\beta} + n^\alpha n_{\alpha;\beta} \right) \\
&= \frac12 \left( n_\alpha n^\alpha_{;\beta} + \left( n^\alpha n_\alpha \right)_{;\beta} - n^\alpha_{;\beta} n_\alpha \right) \\
&= \frac12 \left( -1 \right)_{;\beta} \\
&= 0
\end{align}

\begin{align}
n_{\alpha;\beta} + a_\alpha n_\beta
&= -n_\gamma n^\gamma n_{\alpha;\beta} + n_{\alpha;\gamma} n^\gamma n_\beta \\
&= -n^\gamma \left( n_\gamma n_{\alpha;\beta} - n_\beta n_{\alpha;\gamma} \right) \\
&= -n^\gamma \left( -n_\alpha n_{\beta;\gamma} - n_\beta n_{\gamma;\alpha} + n_\gamma n_{\beta;\alpha} + n_\alpha n_{\gamma;\beta} \right) \\
&= n_\alpha a_\beta + n_{\beta;\alpha}
\end{align}

\begin{align}
K_{\alpha\beta} 
&= -\frac12 \Lie_n \gamma_{\alpha\beta} \\
&= -\frac12 \Lie_n \left( g_{\alpha\beta} + n_\alpha n_\beta \right) \\
&= -\frac12 \left( \Lie_n g_{\alpha\beta} + n_\alpha \Lie_n n_\beta + n_\beta \Lie_n n_\alpha \right) \\
&= -\frac12 \left( n_{\beta;\alpha} + n_{\alpha;\beta} + n_\alpha \left( n_{\beta;\gamma} n^\gamma + n^\gamma_{;\beta} n_\gamma \right) + n_\beta \left( n_{\alpha;\gamma} n^\gamma + n^\gamma_{;\alpha} n_\gamma \right) \right) \\
&= -\frac12 \left( n_{\beta;\alpha} + n_{\alpha;\beta} + n_\alpha n^\gamma n_{\beta;\gamma} + n_\beta n^\gamma n_{\alpha;\gamma} \right) \\
&= -\frac12 \left( n_{\alpha;\beta} + n_{\beta;\alpha} + a_\alpha n_\beta + a_\beta n_\alpha \right) \\
&= -n_{\alpha;\beta} - a_\alpha n_\beta
\end{align}

\begin{align}
\varepsilon_{\alpha\beta\gamma\delta} B^\gamma n^\delta
&= \varepsilon_{\alpha\beta\gamma\delta} \tilde F^{\gamma\epsilon} n_\epsilon n^\delta \\
&= \frac12 \varepsilon_{\alpha\beta\gamma\delta} \varepsilon^{\gamma\epsilon\zeta\eta} F_{\zeta\eta} n_\epsilon n^\delta \\
&= -\frac12 \delta_\alpha^{\left[\gamma\right.} \delta_\beta^\epsilon \delta_\gamma^\zeta \delta_\delta^{\left.\eta\right]} F_{\zeta\eta} n_\epsilon n^\delta \\
&= -\frac12 \left( \delta_\alpha^\epsilon \delta_\beta^\zeta \delta_\delta^\eta + \delta_\alpha^\zeta \delta_\beta^\eta \delta_\delta^\epsilon + \delta_\alpha^\eta \delta_\beta^\epsilon \delta_\delta^\zeta - \delta_\alpha^\eta \delta_\beta^\zeta \delta_\delta^\epsilon - \delta_\alpha^\zeta \delta_\beta^\epsilon \delta_\delta^\eta - \delta_\alpha^\epsilon \delta_\beta^\eta \delta_\delta^\zeta \right) F_{\zeta\eta} n_\epsilon n^\delta \\
&= -\frac12 \left( F_{\beta\delta} n_\alpha + F_{\alpha\beta} n_\delta + F_{\delta\alpha} n_\beta - F_{\beta\alpha} n_\delta - F_{\alpha\delta} n_\beta - F_{\delta\beta} n_\alpha \right) n^\delta \\
&= \left(F_{\beta\alpha} n_\delta + F_{\alpha\delta} n_\beta + F_{\delta\beta} n_\alpha \right) n^\delta \\
&= -F_{\beta\alpha} + E_\alpha n_\beta - E_\beta n_\alpha
\end{align}
\begin{gather}
\Rightarrow F_{\alpha\beta} = n_\alpha E_\beta - n_\beta E_\alpha + \varepsilon_{\alpha\beta\gamma\delta} B^\gamma n^\delta
\end{gather}

\begin{align}
\tilde F_{\alpha\beta} 
&= \frac12 \varepsilon_{\alpha\beta\gamma\delta} F^{\gamma\delta} \\
&= \frac12 \varepsilon_{\alpha\beta\gamma\delta} \left( n^\gamma E^\delta - n^\delta E^\gamma + \varepsilon^{\gamma\delta\epsilon\zeta} B_\epsilon n_\zeta\right) \\
&= \varepsilon_{\alpha\beta\gamma\delta} n^\gamma E^\delta + \frac12 \varepsilon_{\alpha\beta\gamma\delta} \varepsilon^{\gamma\delta\epsilon\zeta} B_\epsilon n_\zeta \\
&= -\varepsilon_{\alpha\beta\gamma\delta} E^\gamma n^\delta - \left( \delta_\alpha^\epsilon \delta_\beta^\zeta - \delta_\alpha^\zeta \delta_\beta^\epsilon \right) B_\epsilon n_\zeta \\
&= -\varepsilon_{\alpha\beta\gamma\delta} E^\gamma n^\delta - B_\alpha n_\beta + B_\beta n_\alpha \\
&= n_\alpha B_\beta + n_\beta B_\alpha - \varepsilon_{\alpha\beta\gamma\delta} E^\gamma n^\delta
\end{align}

\begin{gather}
n_\gamma \varepsilon^{\alpha\beta\gamma\delta} n_\delta = 0 \Rightarrow \varepsilon^{\alpha\beta\gamma\delta} = -\varepsilon^{abc} e^\alpha_a e^\beta_b e^\gamma_c n^\delta \\
\gamma^\alpha_\beta \varepsilon^{\beta\gamma\delta\epsilon} n_\epsilon = \varepsilon^{\alpha\gamma\delta\epsilon} n_\epsilon
\end{gather}

\section{Laws}

\begin{align}
-n_\alpha F^{\alpha\beta}_{;\beta}
&= -n_\alpha \left( n^\alpha E^\beta - n^\beta E^\alpha + \varepsilon^{\alpha\beta\gamma\delta} B_\gamma n_\delta \right)_{;\beta} \\
&= -n_\alpha \left( n^\alpha_{;\beta} E^\beta + n^\alpha E^\beta_{;\beta} - n^\beta_{;\beta} E^\alpha - n^\beta E^\alpha_{;\beta} + \varepsilon^{\alpha\beta\gamma\delta} B_{\gamma;\beta} n_\delta + \varepsilon^{\alpha\beta\gamma\delta} B_\gamma n_{\delta;\beta} \right) \\
&= E^\beta_{;\beta} + n_\alpha n^\beta E^\alpha_{;\beta} - n_\alpha \varepsilon^{\alpha\beta\gamma\delta} B_\gamma a_\delta n_\beta \\
&= g^{\alpha\beta} g_\alpha^\gamma g_\beta^\delta E_{\gamma;\delta} - n_{\alpha;\beta} n^\beta E^\alpha \\
&= g^{\alpha\beta} \gamma_\alpha^\gamma \gamma_\beta^\delta E_{\gamma;\delta} + g^{\alpha\beta} n_\alpha n^\gamma n_\beta n^\delta E_{\gamma;\delta} - a_\alpha E^\alpha \\
&= \gamma^{ab} e^\alpha_a e^\beta_b \gamma_\alpha^\gamma \gamma_\beta^\delta E_{\gamma;\delta} - n^\gamma n^\delta E_{\gamma;\delta} - a_\alpha E^\alpha \\
&= \gamma^{ab} E_{a;b} + n^\gamma_{;\delta} n^\delta E_{\gamma} - a_\alpha E^\alpha \\
&= E^a_{;a} \\
&= -n_\alpha 4\pi j^\alpha \\
&= 4\pi\rho
\end{align}

\begin{align}
-n_\alpha \tilde F^{\alpha\beta}_{;\beta}
&= -n_\alpha \left( n^\alpha B^\beta - n^\beta B^\alpha - \varepsilon^{\alpha\beta\gamma\delta} E_\gamma n_\delta \right)_{;\beta} \\
&= B^a_{;a} \\
&= 0
\end{align}

\begin{align}
\gamma^\alpha_\beta F^{\beta\gamma}_{;\gamma}
&= \gamma^\alpha_\beta \left( n^\beta E^\gamma - n^\gamma E^\beta + \varepsilon^{\beta\gamma\delta\epsilon} B_\delta n_\epsilon \right)_{;\gamma} \\
&= \gamma^\alpha_\beta \left( n^\beta_{;\gamma} E^\gamma + n^\beta E^\gamma_{;\gamma} - n^\gamma_{;\gamma} E^\beta - n^\gamma E^\beta_{;\gamma} + \varepsilon^{\beta\gamma\delta\epsilon} B_{\delta;\gamma} n_\epsilon + \varepsilon^{\beta\gamma\delta\epsilon} B_\delta n_{\epsilon;\gamma} \right) \\
&= \gamma^\alpha_\beta \left( -\left(E^\beta_{;\gamma} n^\gamma - n^\beta_{;\gamma} E^\gamma \right) + K E^\beta + \varepsilon^{\beta\gamma\delta\epsilon} B_{\delta;\gamma} n_\epsilon - \varepsilon^{\beta\gamma\delta\epsilon} B_\delta a_\epsilon n_\gamma \right) \\
&= -\gamma^\alpha_\beta \Lie_n E^\beta + K E^\alpha + \varepsilon^{\alpha\gamma\delta\epsilon} B_{\delta;\gamma} n_\epsilon - \varepsilon^{\alpha\gamma\delta\epsilon} B_\delta a_\epsilon n_\gamma \\
&= -\gamma^\alpha_\beta \Lie_n E^\beta + K E^\alpha + \varepsilon^{\alpha\beta\gamma\delta} B_{\gamma;\beta} n_\delta + \varepsilon^{\alpha\beta\gamma\delta} a_\beta B_\gamma n_\delta \\
&= -e^\alpha_a \dot E^a + K E^a e^\alpha_a + \varepsilon^{abc} e^\alpha_a e^\beta_b e^\gamma_c B_{\gamma;\beta} + \varepsilon^{abc} e^\alpha_a e^\beta_b e^\gamma_c a_\beta B_\gamma \\
&= -e^\alpha_a \dot E^a + K E^a e^\alpha_a + \varepsilon^{abc} e^\alpha_a B_{c;b} + \varepsilon^{abc} e^\alpha_a a_b B_c \\
&= \gamma^\alpha_\beta 4\pi j^\alpha \\
&= 4\pi J^\alpha \\
&= 4\pi J^a e^\alpha_a
\end{align}
\begin{gather}
-\dot E^a + K E^a + \varepsilon^{abc} B_{c;b} + \varepsilon^{abc} a_b B_c = 4\pi J^a
\end{gather}

\begin{align}
\gamma^\alpha_\beta \tilde F^{\beta\gamma}_{;\gamma}
&= \gamma^\alpha_\beta \left( n^\beta B^\gamma - n^\gamma B^\beta - \varepsilon^{\beta\gamma\delta\epsilon} E_\delta n_\epsilon \right)_{;\gamma} \\
&= -e^\alpha_a \dot B^a + K B^a e^\alpha_a - \varepsilon^{abc} e^\alpha_a E_{c;b} - \varepsilon^{abc} e^\alpha_a a_b E_c \\
&= 0
\end{align}

\begin{gather}
\nabla \cdot \ura E = 4\pi \rho \\
\nabla \cdot \ura B = 0 \\
\dot{\ura E} = K \ura E + \left(\nabla + \ura a\right) \times \ura B - 4 \pi \ura J \\
\dot{\ura B} = K \ura B - \left(\nabla + \ura a\right) \times \ura E 
\end{gather}


\section{Modified}

\subsection{Equations}
\begin{gather}
\left(F^{\alpha\beta} + g^{\alpha\beta} \phi\right)_{;\beta} = k n^\alpha \phi + 4\pi j^\alpha \\
\left(\tilde F^{\alpha\beta} + g^{\alpha\beta} \psi\right)_{;\beta} = k n^\alpha \psi
\end{gather}

\begin{align}
-n_\alpha \left( F^{\alpha\beta} + g^{\alpha\beta} \phi \right)_{;\beta}
&= -n_\alpha F^{\alpha\beta}_{;\beta} - n^\beta \phi_{;\beta} \\
&= E^a_{;a} - \Lie_n \phi \\
&= -n_\alpha \left( kn^\alpha \phi + 4\pi j^\alpha \right) \\
&= k\phi + 4\pi\rho
\end{align}
\begin{align}
-n_\alpha \left( \tilde F^{\alpha\beta} + g^{\alpha\beta} \psi \right)_{;\beta}
&= -n_\alpha \tilde F^{\alpha\beta} - n^\beta \psi_{;\beta} \\
&= B^a_{;a} - \Lie_n \psi \\
&= -n_\alpha kn^\alpha \psi \\
&= k\psi
\end{align}

\begin{align}
\gamma^\alpha_\beta \left( F^{\beta\gamma} + g^{\beta\gamma} \phi\right)_{;\gamma} 
&= \gamma^\alpha_\beta F^{\beta\gamma}_{;\gamma} + \gamma^{\alpha\gamma} \phi_{;\gamma} \\
&= e^\alpha_a \left( -\dot E^a + K E^a + \varepsilon^{abc} \left(B_{c;b} + a_b B_c\right) \right) + \gamma^{ac} e^\alpha_a e^\gamma_c \phi_{;\gamma} \\
&= e^\alpha_a \left( -\dot E^a + K E^a + \varepsilon^{abc} \left(B_{c;b} + a_b B_c\right) + \gamma^{ac} \phi_{;c} \right) \\
&= \gamma^\alpha_\beta \left( k n^\beta \phi + 4\pi j^\alpha \right) \\
&= 4\pi J^\alpha \\
&= 4\pi J^a e^\alpha_a
\end{align}

\begin{align}
\gamma^\alpha_\beta \left( \tilde F^{\beta\gamma} + g^{\beta\gamma} \psi\right)_{;\gamma}
&= e^\alpha_a \left( -\dot B^a + K B^a - \varepsilon^{abc} \left( E_{c;b} + a_b E_c \right) + \gamma^{ac} \psi_{;c} \right) \\
&= \gamma^\alpha_\beta k n^\beta \psi \\
&= 0
\end{align}

\begin{gather}
\dot \phi = - k\phi + \nabla \cdot \ura E - 4\pi \rho \\
\dot\psi = -k\psi + \nabla \cdot \ura B \\
\dot{\ura E} = K \ura E + \left(\nabla + \ura a\right) \times \ura B + \nabla\phi - 4 \pi \ura J \\
\dot{\ura B} = K \ura B - \left(\nabla + \ura a\right) \times \ura E + \nabla\psi
\end{gather}


\end{document}